Understanding Satellite Navigation

By Rajat Acharya

This book explains the basic principles of satellite navigation technology with the bare minimum of mathematics and without complex equations. It helps you to conceptualize the underlying theory from first principles, building up your knowledge gradually using practical demonstrations and worked examples. A full range of MATLAB simulations is used to visualize concepts and solve problems, allowing you to see what happens to signals and systems with different configurations. Implementation and applications are discussed, along with some special topics such as Kalman Filter and Ionosphere.

With this book you will learn:

• How a satellite navigation system works
• How to improve your efficiency when working with a satellite navigation system
• How to use MATLAB for simulation, helping to visualize concepts
• Various possible implementation approaches for the technologyThe most significant applications of satellite navigation systems

• Teaches the fundamentals of satellite navigation systems, using MATLAB as a visualization and problem solving tool
• Worked out numerical problems are provided to aid practical understanding
• On-line support provides MATLAB scripts for simulation exercises and MATLAB based solutions, standard algorithms, and PowerPoint slides

• Language: English
• Publisher: Academic Press
• Release date: Aug 19, 2014
• ISBN: 9780128001899

Rajat Acharya

Dr. Rajat Acharya works as a Scientist at the Space Applications Centre, a unit of the Indian Space Research Organisation (ISRO). He is involved with the Indian Satellite Navigation program of GAGAN and IRNSS for more than a decade with pertinent contributions in ionospheric modelling. He is also a Faculty member at the Centre for Space Science and Technology Education – Asia Pacific, where he teaches on the M.Tech course on Satellite Communications and Satellite Navigation. He serves as a visiting Professor at Gujurat University, teaching on the PG Diploma course on Geo-informatics and Satellite Communications. He was also a member of the working group on models and algorithms from ISRO in the second meeting of the International Committee on Global Navigation Satellite System (ICG).

Book preview

Understanding Satellite Navigation

Rajat Acharya

Table of Contents

Cover image

Title page

Copyright

Dedication

Preface

Acknowledgment

Chapter 1. Introduction to Navigation

1.1. Introduction

1.2. Navigation

1.3. Referencing a position

1.4. Radio navigation system

Conceptual questions

Chapter 2. Satellite Navigation

2.1. Satellite navigation

2.2. Architectural components

2.3. Control segment

Conceptual questions

Chapter 3. Satellites in Orbit

3.1. Kepler's laws and orbital dynamics

3.2. Orbital orientation relative to earth

3.3. Perturbation of satellite orbits

3.4. Different types of orbit

3.5. Selection of orbital parameters

Conceptual questions

Chapter 4. Navigation Signals

4.1. Navigation signal

4.2. Navigation data

4.3. Ranging codes

4.4. Encryption

4.5. Multiple access

4.6. Digital modulation

4.7. Typical link calculations

Conceptual questions

Chapter 5. Navigation Receiver

5.1. Navigation receiver

5.2. Functional units of user receivers

Conceptual questions

Chapter 6. Navigation Solutions

6.1. Fundamental concepts

6.2. Generation of observation equation

6.3. Linearization

6.4. Solving for position

6.5. Other methods for position fixing

6.6. Velocity estimation

Conceptual questions

Chapter 7. Errors and Error Corrections

7.1. Scope of errors

7.2. Control segment errors

7.3. Space segment errors

7.4. Propagation and user segment errors

7.5. Techniques of error mitigation

7.6. Effect of errors on positioning

7.7. Error budget and performances

Conceptual questions

Chapter 8. Differential Positioning

8.1. Differential positioning

8.2. Differential correction techniques

8.3. Implementation of differential systems

Conceptual questions

Chapter 9. Special Topics

9.1. Kalman filter

9.2. The ionosphere

Conceptual questions

Chapter 10. Applications

10.1. Introduction

10.2. Applications overview

10.3. Specific applications

Appendix 1. Satellite Navigational Systems

Index

Copyright

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

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ISBN: 978-0-12-799949-4

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Dedication

Dedicated to the loving memory of Chiku, Charki and Gunda

Preface

I was very much motivated to write a book on Satellite Navigation Principles when my booklet published internally in ISRO on this subject for the Satellite Navigation Course by the Centre for Space Science and Technology Education-Asia Pacific (CSSTE-AP) was much appreciated by my students. From the feedback I received from people from a wide spectrum of professions and countries and represent both commercial and strategic users, as well as researchers, it was clear that a book was needed which was acceptable equally in terms of ease of understanding and pertinent. It was obvious that a good introductory book with a selective assortment of subjects and explanations simple enough for new learners was necessary. So, this book has been written with the aim of introducing the subject to beginners, coming from different backgrounds, in a very congenial form, so that the subject can be learned from scratch.

One of the main features of this book is that it explains the basic working principles right from a foundation level with just the necessary mathematics and no complex-looking equations. This book is written keeping in mind the undergraduates and all those readers who are being introduced to the subject for the first time. It is expected that the simplistic approach developed from the first principle yet incorporating all technicalities, will not only serve them well but also be a source of enjoyable learning.

This book will cover the basic working principles of the generic satellite navigation system, instead of concentrating on any particular existing system. It will emphasize the build up of fundamental ideas for each involved process based on elementary physics added to rational common sense. Every basic principle is followed by mathematical substantiation, but using only as much mathematics as is deemed necessary for target readers. In addition, this book employs MATLAB as a visualisation tool, for every important new concept that is introduced. This will allow readers to corroborate all that they have learned through simulation, which we believe makes this book unique.

Finally, it is worth mentioning that all the relevant topics on the subject have been encompassed in a comprehensive manner. However the views and the opinions presented here are those of the author and do not necessarily reflect the views of his employer, nor the Government of India.

Rajat Acharya

Acknowledgment

I wish to extend my sincere gratitude to the University of Calcutta and especially to personalities like Prof. Asish Dasgupta, Prof. Apurba Datta and Prof. Bijoy Banerjee, who have always been my inspiration. I am also grateful to the, Space Applications Centre (SAC-ISRO) for the support it has provided me. The courtesy extended by Dr Bijoy Roy, Dr Chandrashekhar, Mr Suman Aich, and Mr Ananya Roy of SAC by agreeing to review and comment on the early versions of the manuscript are also very much appreciated. Thanks are due to Dr Suman Ganguly of CFRSI for his kind cooperation. I also acknowledge the generous gestures of Dr M R Sivaraman, Dr Kalyan Bandyopadhyay, Mr Vilas Palsule and the whole team of the CSSTE-AP. I would also like to thank the entire Elsevier team and the peer reviewers of this book. Last but not the least, I sincerely thank my wife Chandrani, my son Anubrata, and my parents, for their inspiration and for the relentless sacrifices they made during the writing of this book in order to make this effort a success.

Chapter 1

Introduction to Navigation

Abstract

Chapter 1 introduces the reader to the generic term navigation and introduces its subtleties with relevance to everyday life. It starts with the formal definition of the term followed by the history of navigation from its prehistoric beginnings to the current state of the art. The development of navigation techniques over the years is described in a concise manner. It then explains the different forms of navigation with a brief description of one system for each of these kinds. Prerequisite topics of the reference frame and coordinate system are also discussed and elaborated on with reference to the geodetic shape of the earth.

Keywords

Celestial navigation; Datum; Dead reckoning; ECEF frame; ECI frame; Ellipsoid; Geoid; Guidance; Inertial navigation system; Instrumental landing system; LORAN; navigation; Orthometric height; Piloting

Chapter Outline

1.1 Introduction 1

1.1.1 Organization of this book 2

1.2 Navigation 4

1.2.1 History of navigation 4

1.2.2 Types of navigation 8

1.2.2.1 Guidance 8

1.2.2.2 Dead reckoning 8

1.2.2.3 Piloting 9

1.3 Referencing a position 9

1.3.1 Reference frame 11

1.3.1.1 Heliocentric reference frame 12

1.3.1.2 Geocentric reference frames 12

1.3.1.3 Local reference frames 17

1.3.1.4 Conversions between coordinate systems 17

1.4 Radio navigation system 21

1.4.1 Piloting system 22

1.4.2 Guidance system 23

1.4.3 Dead reckoning system 24

Conceptual questions 25

References 25

1.1. Introduction

Navigation is a basic need for anyone who wants to move with a purpose. Navigation is the art of moving in a suitable direction in order to arrive at a desired location. Thus, even in prehistoric times, when the most primitive form of animals started moving on earth, the art of navigation existed in its most ancient form. Even today, when humans, the most evolved species on earth, move by flying in the most technologically advanced aircraft or by driving a car, or by riding a bicycle or simply walking, with a desire to reach somewhere, we perform some sort of navigation.

You may have noticed that when we move without the aid of instruments and the route to our destination is known to us, we generally use some sort of mental map, which is mostly pictorial in the form of landmarks and connected paths. On this map, we identify our positions and apply our previous experience to guide us and decide the course of our movement. However, this method does not work for a new destination or for places where such landmarks are not present, which is the reason why people get lost in deserts or on the oceans. In such situations, we paper or digital maps, which give similar information. However, whether paper or digital, or as mental pictures including other geographical information, these maps are aids to navigation that enable us to locate and relate our positions with respect to our destinations and show different possible ways to reach there. The decisions we make in choosing the course of our movement by comparing our position with the available information on these maps is called navigation.

Thus, it is apparent that we first need to know our position to identify correctly where we are, and then to make an appropriate decision about where to move. Satellite navigation is a method that provides us with the correct position on or off the earth for this purpose. Here, signals transmitted from navigation satellites are used to derive the required set of position parameters by a navigation receiver. In turn and in conjunction with the additional information, these parameters are used to further decide the course of movement.

However, positions are not sought only for movement. Sometimes our exact position is also required to be correlated with other facts or to derive ancillary information. For example, if we know our position on the earth's surface, we can easily figure out the kind of climate we must expect. Knowing precise positions of a network of points on the earth will also let one obtain the exact shape of the earth or its derivatives, such as tectonic or other crustal movements. There are many other interesting applications of navigation, which we will discuss in Chapter 10. There, we shall come to know how this knowledge about position and its derivatives can be used for many exciting applications.

The general requirement of the estimation of position is global; for that, we need to represent positions uniquely. Positions are hence represented in terms of global standards such that positions of all the points on and near the earth can be expressed by a certain unique coordinate based on a common reference. It is like the unique identity of that position. Thus, finding the position of a person is simply a matter of determining the unique identity of the place where he or she is currently located. These coordinates are hence chosen to specify the positions in a convenient manner. In later subsections of this chapter, we will learn about reference frames and coordinate systems, which forms the basis for representing the positions. Nevertheless, the definition of these coordinates assumes the existence of certain geodetic parameters.

1.1.1. Organization of this book

The philosopher Socrates said Know thyself. At the outset of learning navigation, we can update this to say, Know (the position of) thyself. Thus, our entire endeavor throughout this book will be to understand the fundamentals of how modern space technology is used to fix our own position, aided by advanced techniques and effective resources. Details about existing systems currently being used for this purpose will be discussed post hoc.

However, it is also important to know how the information is organized in this book. The more logically things are developed here, the more easy it will be to understand them. Thus, it is a good idea to first have a holistic view of how the different aspects of a satellite navigation system are gradually introduced in the chapters in this book. We therefore suggest that readers continue to pursue this section describing the overall organization of this book, about which many of us have a general apathy and a tendency to want to skip this explanatory material.

The first chapter of this book is informative. We will start by introducing the term ‘navigation’ and getting a feel for the real development of a navigation system through a chronological description from their inception up to the current state of the art. We will first learn about the historical development of the navigation system. Whilst to some history may sound boring according to Sir Francis Bacon Histories make men wise. We will therefore take a look at the history of satellite navigation before we gear up to understand the technological aspects of the subject. Then, before we move on to the topics of satellite-based navigation, a brief introduction to its predecessors, including other forms of navigation, should prove helpful. All of these will be covered in this chapter, and reading it, we hope, will be as interesting as the technology in subsequent chapters. Chapter 2 is also information based, primarily regarding the overall architectural segments of the whole satellite navigation system. Although we will only learn in detail about the control segment in this chapter, other elements will be discussed in the following chapters. Enjoyment of this book will intensify in Chapter 3, where we describe the space segment of the architecture. From this chapter onward, there will be frequent Matlab activities illustrating the current topic. We suggest that readers attempt these activities as they come across them, rather than leaving them to the end. Chapter 4 details the satellite signals used for navigation purposes and transmitted by satellites. Their characteristics will be described and the rationale for their use explained. Chapter 5 describes the user segment and will provide the working principles of a navigation receiver and the different aspects of it. We will explain how signals are used in receivers to derive the parameters required to fix a position. Chapter 6 explains the algorithms for the derivation of the navigation parameters i.e. position, velocity, and time, by using the measurements and estimations performed in the receivers. Receiver errors in such estimations with their sources and effects are discussed in detail in Chapter 7. Chapter 8 contains the topic of differential navigation system. It is a vast subject that could easily fill a book the same size as this or even bigger. However, we have accommodated it here into a single concise chapter of only few pages. Chapter 9 looks at special topics such as the Kalman filter and the ionosphere, both of which have large implications for navigation systems. Readers may skip reading this particular chapter if they wish, without loss of continuity. However, that would be at the cost of some very interesting material. Finally, Chapter 10 provides details of some important applications of satellite navigation.

1.2. Navigation

Navigation is related to the art of getting from one place to another, safely and efficiently. Although, the word ‘navigation’ stems from the Latin word Navigare, which means ‘to sail or drive a ship,’ its contemporary meaning is the art of providing position and/or direction to anyone on land or sea or in space (Elliot et al. 2001).

1.2.1. History of navigation

The art of navigation predated the advent of mankind. Prehistoric animals moved in search of food using their innate navigation skills. Figure 1.1, however, is only indicative.

Humans have been using different techniques of navigation from the early ages of civilization. Primitive people living in caves had to hunt deep in the forest in search of food when geographical movement was not easy and finding their way back was difficult. Thus, they made special marks on trees or erected stone pillars to create landmarks in order to find their way back home. The use of sound or smoke signals were another common means of finding their way back. We will look more at the formal classifications later, but it is worth mentioning here that these most primitive methods of navigation were of the guidance type.

Navigation developed at sea, with most developments in the modern navigation system occurring in the process of guiding sea vehicles. In ancient times, seafaring explorers started traveling across the oceans in search of new lands, in order to increase trade and colonize. Development in the field became necessary in order to cater to the needs of voyages and the constant effort to improve them. As mentioned before, mapping the sea along side its adjacent lands is not navigation, but deriving ones own position from it and thus deciding the direction of movement toward a certain destination is. However, both systems were developed simultaneously and are sometimes treated as the same thing. In this section, we restrict our discussion to navigation only.

FIGURE 1.1   Primitive navigation.

The first kind of sea navigating was probably done by skirting around the coast, and thus by staying in sight of land. Pictorial maps were created during this time by sailors who would draw what they could see along the coast. Using these, they could return or retrace their course on subsequent journeys. The first known coastal and river maps were from China in around 2000 B.C. and indicated sailing directions (Wellenhof et al. 2003). However, when voyages ventured further out into the sea, the only means of navigating was by observing the position of the sun and stars. This kind of navigation is termed celestial navigation. Some experienced sailors could also navigate by understanding the winds or determining the depth of the seabed, from which they could estimate their distance from the land. This was probably the earliest form of bathymetric navigation.

Written records of celestial navigation date back to the third century B.C. Some of these accounts are available in Homer's epic, The Odyssey (History of Navigation, 2007). The astrolabe, which measures the elevation of the sun and the stars, as shown in Figure 1.2(a) became the main instrument for positioning and was apparently used even before 600 BC (Kayton, 1989). Heron and Vitruvious gave a detailed description of the odometer, an instrument to measure distance (Wellenhof et al. 2003). During this time, Greek and Egyptian sailors started using the polar stars and constellations to navigate because they did not disappear below the horizon throughout the night. However, movement using polestars needed to be corrected with time as the stars change their positions because of the wobbling of the earth on its axis. Measurements of the instruments were aided by nautical charts. Ptolemy produced the first world map, which remained in use for many years during sea voyages. Textual descriptions for sailing directions have been in use in one form or another since then.

The middle ages in navigation were marked by the discovery of lodestone. With this, navigation became easier for sailors, who started to use it for its magnetic properties. Comparing it with detailed maps of the period, they could find their way easily even with unfavorable sky conditions allowing sailors to navigate even with limited visibility. The first true mariner's compass was invented in Europe toward the beginning of the thirteenth century A.D. Thus, when Christopher Columbus set out on his transatlantic voyage in 1492, he had only a compass, a few dated measuring instruments, a method to correct for the altitude of Polaris, and some rudimentary nautical charts as tools for navigation.

From the middle of the sixteenth century, navigation saw a rapid development in related technology when a number of instruments and methods were invented. This was when the Europeans started to settle colonies in different countries, and they used sea routes to navigate to these new lands. The improvement of navigational techniques became mandatory and the mathematical approach toward navigation made it a scientific discipline. By the seventeenth century, the quadrant had become one of the dominant instruments. Magnetic variations were studied and the magnetic dip, the angular inclination of the geomagnetic lines of force at a location, was discovered, which gave enormous support to position finding. The defined nautical mile could also now be measured with much accuracy. By the middle of the eighteenth century, the invention of instruments such as the sextant and the chronometer, marked the onset of modern times in navigation. The sextant, as shown in Figure 1.2(b), could measure the elevation of the sun, moon or a star by aligning its reflection from a semi-reflecting surface with the visible horizon seen directly through it. In the process, its own relative position could be estimated when the position of the star or the sun was known.

FIGURE 1.2   (a) Working principle of Astrolabe, (b) Sextant.

Time always remained an important parameter when finding a position. The sundial was a primitive clock, the oldest of which was found in Egypt at Karnak. It was also used in ancient Greece and China (Kayton, 1989). Later, the pendulum clock was invented and was used to keep time on land but was not suitable for marine platforms. Thus, for the seagoing vehicles and mariners, the best devices were still water and sand clocks. This put a serious constraint on the accurate determination of longitude, which needs a precise knowledge of time. Around the middle of the eighteenth century, a huge sum of prize money was offered to anyone who could provide a precise method of measuring longitude. In 1759, John Harrison invented a clock that was accurate within a few seconds over a period of around 6 months. Captain Cook used the Harrison Clock for his expedition to the Antarctic. Another remarkable event that took place in the determination of longitude was the landmark decision of adopting the prime meridian (0° longitude) in 1884. It remains the basis of positioning even today.

In the last decade of the nineteenth century, radio communications started in the form of wireless telegraphs. For sea goers, signals started being sent to ships not only in the form of messages, but to allow navigators to correct their chronometers. Radio communication between ships also helped sailors to make navigational decisions.

Radio-based navigation systems advanced rapidly during World War II. By this time, the quartz clock became available and microwaves were used extensively with navigational devices and British physicist Robert Watson first demonstrated the Radio Detection and Ranging system (RADAR) as a warning system against air attacks. This technology was readily implemented in ships as a navigation aid. Soon after this, Alfred Loomis suggested a radio-based electronic navigation system, which was later developed into the Long Range Navigation (LORAN) system.

Radio navigation was ushered into a new era in October 1957, when the former Soviet Union (USSR) launched the world's first artificial satellite, Sputnik. Scientists used the Doppler shift of the Sputnik’s signal to obtain the satellite's position and velocity. Subsequently, a series of satellite-based navigation programs had been undertaken and established by both the United States and the USSR. During this time, satellite constellations for navigation based on both Doppler and ranging came into use. The TRANSIT satellites that started operating in 1964 were Doppler-based navigation systems, whereas the SECOR system was based on ranging. These were followed by the Russian TSIKADA and American TIMATION systems. TIMATION was planned for time transfer by sending precise clocks into space. The results of these precursor programs formed the basis of today's Global Positioning System (GPS) of NAVSTAR (Parkinson, 1996) and GLONASS. Currently, many countries and groups of nations use their own satellite-based navigation system: the Galileo system in the European Union, COMPASS in China, and IRNSS in India are some of them. We shall learn about the basic working principles of these systems in later chapters.

1.2.2. Types of navigation

A modern navigation system is typically a radio navigation system that is non-autonomous in nature. It means, the system operates only when an appropriate external signal is received by a receiver (Wellenhof et al., 2003). It provides position, velocity, and time (PVT) in a three-dimensional system. However, there are certain systems which only give the path that one should follow to reach the destination. Based on the nature of parameters and how they are derived, modern navigation systems can be divided into three broad types.

1.2.2.1. Guidance

Guidance is a type of navigation that provides only a course to a destination for the user, but with no information about its exact position. Thus, the user only knows the exact route that should be followed to lead him or her to his destination, with no knowledge of his or her present position.

Guidance is the oldest type of navigation. The movements of early travelers finding their way to their destination by observing the rising and setting of the sun and the moon and orientation using constellations were navigation of the guidance type. In modern times, when you follow the markers in a big airport directing you to reach your designated terminal, or when you decide your route on the basis of displayed signs on a highway, this is navigation of the guidance type. Thus, we frequently utilize guidance navigation throughout our lives sometimes without even realizing it.

Some modern radio navigation systems are in this group, such as the instrument landing system (ILS) and microwave landing system, which are used for aircraft.

1.2.2.2. Dead reckoning

Sometimes it is difficult to use guidance navigation, especially for long-range movement. In such cases, it is more convenient to know one's current position rather than be guided from origin to destination. But how do we update our position with time? Position at the current instant can be determined from the positions at any prior time using the value of time elapsed since then and some simple dynamic parameters obtained as position derivatives. Thus, current positions of any moving entity can be deduced in relation to any prior position, or even with respect to the point of its origin of movement. One of the easiest methods of doing this is by using the basic principles of Newton's laws of dynamics. From these laws, we derive the current position and velocity of the body as

(1.1)

(1.2)

where Sk and Sk−1 are the positions at time tk and its previous instant tk−1, respectively. vk and vk−1 are the corresponding velocities, and fk−1 is the acceleration at instant tk−1.

Therefore, the position Sk and velocity vk at any instant tk may be derived from those at its previous instant tk−1, just by knowing the acceleration, fk−1, and from their previous values Sk−1 and vk−1, respectively. Similarly, the values of Sk−1 and vk−1 can be derived from parameters at an instant even before them. Therefore, extending this logic backward, we can say that if we know the position S0 at any starting point with a standstill condition, i.e. (v0  =  0) at any time instant t0, we can find out its position and velocity at any later time tk just by measuring acceleration f at the initial instant and all intermediate instants. Because the present position is deduced from an initial standstill condition (i.e. the ‘dead’ condition of the body) from which we start reckoning (i.e. calculating the position), this kind of navigation is called dead reckoning and the term is also sometimes said to be derived from the word ‘deduced’ (Meloney, 1985).

There are six degrees of freedom for any massive rigid body. Degrees of freedom are the set of independent dimensions of motion of the rigid body that completely specify the movement and orientation of the body in space. They indicate the independent directions in which a body can exhibit linear or rotational motion without affecting its similar movement in any other direction. In accounting for such motions, the three directions of translational motion defining the position of the body are considered, along with the three orthogonal rotational directions of the body that give its orientation. Thus, in addition to position, dead reckoning can also be used to find the orientation of the body by measuring the angular acceleration of the body about these three rotational axes.

Navigational systems based on these inertial properties of a dynamic system belong to this category, such as the inertial navigation system (INS). Most commercial aircraft use this as their primary navigational system.

1.2.2.3. Piloting

In the piloting or pilotage type of navigation, the user derives navigational parameters (PVT) which are updated each time. New measurements are performed over the update interval, which leads to new positions for every update.

Satellite-based navigation, which is the subject of this book, belongs to this category. Other systems of this kind are hyperbolic terrestrial radio positioning systems such as LORAN etc. (LORAN, 2001; Loran, 2011)

1.3. Referencing a position

We fix our positions using different navigation systems. But the question is, with respect to what? A more fundamental question that may arise is, do we always need to represent our position with respect to something? If yes, then what should that ‘something’ be?

To answer these questions, let us start by using a simple analogy. How do we typically communicate our positions in everyday life? Verbally, we actually tell our location to someone in a fashion such as, I am on Parallel Boulevard, about a 100 yards right from the old lighthouse, or I am at Copley Plaza, about 50  m south of the city library, or I have come across the airport by half a kilometer due east. Notice the common features we use in these statements. In all cases, we refer to our position in terms of distance with respect to some specific reference landmarks such as the lighthouse, the city library, or the airport. Furthermore, we mention the distance from them in definite directions. We also assume that the person to whom the position is being described already knows these landmarks used as references. The description is useless if he or she is new to these places and does not know where the references are. Thus, what we deduce from these are that we need a fixed (or apparently fixed) and defined reference to describe our position, and a distance from or direction away from these reference points. Fixed and known references, defined directions, and definite distances are thus the elements required to describe positions.

The reference used should be universally accepted and understood and should be convenient for referring to any position of interest. We have talked about specific landmarks as references in our example. These references are local and cannot be used to describe positions of any location across the globe. Therefore, to perform pragmatic position fixing, what should be the nature of these references? The obvious answer is that the reference point itself must be located with approximate equal nearness to all points whose positions are to be described so that any point in question may be represented with equal convenience.

Then, once the reference is set, the next requirement is to represent the distances of our position in the best possible way. From any pragmatic reference point O, as in Figure 1.3, there will be a shortest radial range, moving along which the point in question, P, may be reached. The distance is shown as R. However, in our three-dimensional space, this range may be in any arbitrary direction from the reference, O. Describing any arbitrary direction is impossible unless we make use of some predefined standard direction for specifying it. In a three-dimensional space, there can be a maximum of three mutually orthogonal directions. We may fix three such directions in space, each referred to as an axis. The direction of radial range may be described by the angle it makes with such directions. The angles are represented by α, β and γ which are such that the relation cos²  α  +  cos²  β  +  cos²  γ  =  1 is always maintained. The same point P may also be reached by moving vectorial distances along these axes. These paths are nothing but projections of the radial vector distance R on the defined axes shown as a1, a2, and a3 in Figure 1.3. Thus, we can express vector R as

FIGURE 1.3   Orthogonal bases in a three-dimensional space.

(1.3)

are the unit vectors along these three axes and form the basis.

In general, we need to move three orthogonal vectorial distances along the axes to move effectively from the reference point to reach any other point of choice. Any mutually orthogonal vectors can be used to represent the axes for movement, but depending on the need, we use some predefined fixed directions of these vectors because it serves no purpose if these three directions are always chosen arbitrarily.

So, represent the position of a point, we first need to fix a reference point. Then, with respect to this reference, the positions of all other points in question may be described in terms of distances along three fixed, predefined orthogonal axes.

1.3.1. Reference frame

Reference frame: Any arbitrary reference point and associated definition of three orthogonal axes, with respect to which the position of all other points may be defined, constitutes a reference frame. It is typically defined by specifying the position of the reference point and direction of the axes. The reference point is specified by attaching it to any physical system and is referred to as the origin of the frame.

Accordingly, there may be two types of reference frames, described below.

Inertial: An inertial frame of reference is one that is not itself accelerating, and hence one in which the laws of inertia are valid.

Non-inertial: A non-inertial frame of reference is one that is itself accelerating, and hence the laws of inertia are not valid.

We mentioned that the positions of other points are defined in terms of distances along the defined directions from this reference point. The unit vectors along these three orthogonal vectors thus form the basis for describing distances with respect to the reference. There can be different orthogonal sets of basis vectors, and each such set constitutes a coordinate system.

Coordinate System: A defined set of three orthogonal basis vectors associated with each axis of a reference frame, in which the position of any point in space may be described in terms of distances along the axes from the origin.

Different types of coordinate systems, such as Cartesian, spherical, and cylindrical, are used for different purposes. However, for navigation, geodetic reference frames with cartesian or spherical coordinate systems are typically used, as mentioned previously.

1.3.1.1. Heliocentric reference frame

Helios was the Greek sun god whose name was later Latinized as Helius to represent the sun. Thus, from the name, it is evident that heliocentric reference frames are those in which the origin of the reference frame is fixed at the center of the sun. These references are used to represent the positions of the celestial bodies or the positional elements in the solar system. However, it is not suitable for representations of positions over the earth or near it.

1.3.1.2. Geocentric reference frames

We have seen that the location of points in three-dimensional space are most conveniently described by coordinates with an origin around the points. Therefore, to suitably represent positions on the earth and around it, the chosen reference frames are typically geocentric. The origin coincides with the center of the earth and the axes align with the earth's conventional axes or planes. Geocentric reference frames can be naturally divided into different classes, as described subsequently.

To represent positions on the earth and its surroundings, a geocentric reference frame may be defined with Cartesian coordinates. But how are the axes of this frame defined? For an inertial system, the axes should not linearly accelerate or rotate, because rotation is always accompanied by acceleration.

We know that the earth is spinning about its own axis and is revolving around the sun as well. Thus, what appears to us to be fixed and stationary on the earth when we look at it standing on the earth, is not actually so. We can find that everything on the earth is rotating with it, if we look at it from space. Thus, no frame fixed with the earth can be stationary. Furthermore, as the earth revolves round the sun, so too is everything that appears to be fixed upon it.

Then what can we use as a stationary reference frame? Honestly speaking, nothing has yet been found that can be treated as absolutely stationary. No reference may be truly considered inertial. Therefore, we use relative stationarity, or those references that are approximately stationary, compared with the motion of the earth. The distant stars can be used for this purpose. When we look toward the sky from the earth, the distant stars appear to surround us in all directions. These stars apparently remain fixed at their positions throughout the year, whatever the position of the earth around the sun. These stars are at such a great distance that the range of the earth's movement is proportionately negligible compared to it. These distant stars may be assumed to form another hollow sphere of infinite radius concentric with the earth, called the celestial sphere. The deviations of these stars, as observed over the entire movement of the earth, are trivial, and hence can be considered stationary. However, the sun appears to move in this geocentric frame on an elliptical path called the ecliptic. The direction toward the distant stars on the celestial sphere from the geocenter through the point where the ecliptic crosses the equatorial plane with the northward motion of the sun is called the direction of the vernal equinox or the first point of Aries. The angular distances of the distant stars as observed from the earth are reckoned with respect to this fixed direction that marks the beginning of the Aries constellation (Figure 1.4).

FIGURE 1.4   (a) ECI reference frame, (b) Earth-centered, earth-fixed reference frame.

1.3.1.2.1. Earth-centered inertial

The earth-centered inertial (ECI) is a geocentric reference frame that does not rotate with the earth, and hence, the axes retain their orientation relative to these fixed distant stars.

In the ECI frame, as illustrated in Figure 1.4(a), the origin of the frame is at the center of the earth, and the three axes of right-handed orthogonal frame are attached to it. The X and Y axes of the Cartesian coordinates remain on the equatorial plane of the earth mutually perpendicular to each other. The X axis remains always directed toward the first point of Aries, whatever the position of the earth. The Z axis remains aligned with the mean rotational axis of the earth and points toward the North Pole. Because of this, although it is geocentric, the reference frame does not rotate as the earth rotates about its axis. Thus, the coordinate system has a fixed orientation in space, and hence may be called an inertial system as long as other perturbations of the axes are considered trivial. However, the perturbations cannot be totally neglected for precise applications. To alleviate this problem, the directions are defined at a fixed epoch, i.e. the direction that these axes made on January 1, 2000, is taken as the standard (Kaplan et al. 2006). Satellites revolving around the earth experience gravitational pull effective from the point of its mean center but are not affected by the rotation of the earth. Therefore, to represent the positions of satellites revolving around the earth, ECI acts as a suitable reference point. Because the earth rotates whereas the frame does not, the position of the points fixed on the earth’s surface changes with time in this reference frame.

1.3.1.2.2. Earth-centered, earth-fixed

We saw in the last section that in an ECI frame the position of locations fixed on the earth's surface changes with time. This is inconvenient for conventional positioning uses. Thus, if we define a geocentric reference frame in which the axes are fixed with the earth and rotate with it, the coordinate of any position on the earth remains fixed over time. Hence, this problem can be avoided.

Earth-centered, earth-fixed (ECEF) is a geocentric reference system in which the axes are attached to the solid body of the earth and rotate with it. This is shown in Figure 1.4(b). In this kind of system, the origin of the frame is again at the center of the earth and a right-handed frame of axes is attached to it to. The distances are primarily represented in Cartesian coordinates, XYZ. The X and Y axes remain perpendicular to each other. The X axis is fixed along the prime meridian (i.e. 0° longitude) and the Y axis is hence accordingly placed on the equatorial plane along 90°  E longitude. The Z axis remains pointing toward the North